Accuracy of computational solvation free energies for neutral and ionic

compounds: Dependence on level of theory and solvent model

Sierra Rayne a,* and Kaya Forest b

a Ecologica Research, 301-1965 Pandosy Street, Kelowna, British Columbia, Canada V1Y 1R9

b Department of Chemistry, 583 Duncan Avenue West, Okanagan College, Penticton, British Columbia,

Canada V2A 8E1

* Corresponding author. Tel.: +1 250 487 0166. E-mail address: rayne.sierra@gmail.com (S. Rayne).

1

Abstract

Gas to aqueous phase standard state (1 atm→1 mol/L; 298.15 K) free energies of solvation (ΔG°solv)

were calculated for a range of neutral and ionic inorganic and organic compounds using various levels

and combinations of Hartree-Fock and density functional theory (DFT) and composite methods (CBS-

Q//B3, G4MP2, and G4) with the IEFPCM-UFF, CPCM, and SMD solvation models in Gaussian 09

(G09). For a subset of highly polar and generally polyfunctional neutral organic compounds previously

identified as problematic for prior solvation models, we find significantly reduced ΔG°solv errors using

the revised solvent models in G09. The use of composite methods for these compounds also

substantially reduces their apparent ΔG°solv errors. In contrast, no general level of theory effects

between the B3LYP/6-31+G** and G4 methods were observed on a suite of simpler neutral, anionic,

and cationic molecules commonly used to benchmark solvation models. Further investigations on

mono- and polyhalogenated short chain alkanes and alkenes and other possibly difficult functional

groups also revealed significant ΔG°solv error reductions by increasing the level of theory from DFT to

G4. Future solvent model benchmarking efforts should include high level composite method

calculations to allow better discrimination of potential error sources between the levels of theory and

the solvation models.

Keywords: Solvation free energy; Solvation models; Hartree-Fock; Density functional theory;

Composite methods

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The continuing development of implicit solvent models for predicting solvation free energies (ΔGsolv) is

driven by their importance in better understanding and estimating reaction rates, mechanisms, and

equilibria in solution, partitioning in biological, environmental, and engineered systems, and

fundamental aspects of biological and medicinal chemistry.[1, 2] In recent work, a set of 54 highly polar

and generally polyfunctional organic compounds with available experimental solvation free energies

was proposed against which current and future solvation models should be benchmarked.[3] Using

various self-consistent reaction field (SCRF) based solvation models (PCM, CPCM, DPCM, IEFPCM,

IPCM, and SCIPCM) in Gaussian 03 (G03) and Gaussian 98 (G98), relatively poor ΔGsolv prediction

performance was reported for the models, notably with halogenated and heteratom substituted

compounds. However, many of the proposed compounds contain substantial conformational

complexity, resulting in significant work to ensure that the lowest energy gas and aqueous phase

conformers (which could differ from each other) are found when evaluating all potential combinations

of energetic contributions from low-lying states towards the net solvation free energy. In addition, some

compounds are hydrolyzable (e.g., methanesulfonyl chloride; carboxylate, phosphate, and sulfate

esters), which could preclude accurate assessment of their experimental solvation energies. Numerous

other members contain aliphatic and aromatic amines (e.g., imidazole), which can have relevant

acid/base behavior at near neutral aqueous pH values, further complicating reliable experimental

solvation energy measurements against which to assess theoretical estimates.

From this dataset of 54 compounds, we identified a subset of 9 compounds (Table 1) that maintain a

reasonable degree of polyfunctionality and polarity, are relatively constrained with regard to low

energy conformational freedom, lack potentially confounding acid/base behavior in near neutral

aqueous solutions that solvation models attempt to mimic, and which display approximately the same

error distribution as the parent dataset. For example, at the HF/6-31+G** and B3LYP/6-31+G** levels

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of theory [4-12] with the parent n=54 dataset, Guthrie and Povar [3] found root mean squared deviations

(RMSDs) from experimental gas to aqueous phase free energies of solvation (ΔG°solv; denoting transfer

of solute at 298.15 from 1 atm in the ideal gas phase to 1 mol/L in the ideal dilute solution aqueous

phase) of 2.13 (IEFPCM-UAHF/HF/6-31+G**), 2.31 (IEFPCM-UAHF/B3LYP/6-31+G**), 2.09

(CPCM/HF/6-31+G**), and 2.02 kcal/mol (CPCM/B3LYP/6-31+G**) using the IEFPCM [13] and

CPCM [14, 15] solvation models in G03.[16] These authors state they typically used the default settings

with the UAHF radii for specifying the IEFPCM molecular cavity in G03 and G98; the default PCM

radii has changed to UFF in Gaussian 09 (G09; http://www.gaussian.com/g_tech/g_ur/k_scrf.htm). The

corresponding ΔG°solv RMSDs for the 9 compound subset using these two levels of theory and two

solvent models from Guthrie and Povar [3] are 3.1, 3.2, 3.1, and 3.1 kcal/mol, respectively. Our subset

thus does not discriminate against difficult compounds in the parent dataset based on the error metrics

given in Guthrie and Povar;[3] arguably, it concentrates the relative presence of difficult compounds.

Our subset does omit some functional group classes discussed above due to uncertainty regarding the

quality of the experimental ΔG°solv data being benchmarked against, and issues over conformational

complexity and aqueous reactivity. The ΔG°solv span is 10.8 kcal/mol in the n=9 subset versus 13

kcal/mol in the original 54 compound dataset.

Our first investigation was to examine whether changes in the default IEFPCM [13] and CPCM [14, 15]

solvent models from G03 [16] to G09 [17] resulted in different ΔG°solv prediction accuracies at the HF/6-

31+G** and B3LYP/6-31+G** levels of theory [4-12] for these 9 compounds (Table 1). Calculated

ΔG*solv (transfer of solute at 298.15 from 1 mol/L in the ideal gas phase to 1 mol/L in the ideal dilute

solution aqueous phase) were converted to ΔG°solv via the relationship, ΔG°solv = ΔG*solv + RT ln ( TȒ )

where the term RT ln ( TȒ ) is about 1.9 kcal/mol at 298.15 K. All calculations were conducted using

G09,[17] only the lowest energy conformation was considered, no imaginary frequencies were present,

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free energies include both zero-point and thermal corrections, and geometries were optimized in the

respective phases of interest. For all sets of data (both G03 [from ref. [3]] and G09 [present study]), the

IEFPCM and CPCM solvent models yield effectively equivalent ΔG°solv and associated error metrics.

The G09 versions of IEFPCM-UFF and CPCM also give substantially lower ΔG°solv errors than their

G03 default counterparts, with reductions in MAD and RMSD by up to 50%. For our IEFPCM-

UFF/B3LYP/6-31+G** calculations using G09, the MAD and RMSD are near chemical accuracy at 1.2

and 1.4 kcal/mol, respectively, with a maximum individual absolute error (MIAE) of 1.7 kcal/mol,

reduced from the corresponding G03 data at the IEFPCM-UAHF/B3LYP/6-31+G** level from ref. [3]

of 2.2 (MAD), 3.1 (RMSD), and 7.5 (MIAE) kcal/mol, respectively. Based on the updated error metrics

at the IEFPCM-UFF/B3LYP/6-31+G** level in G09, one would no longer consider these particularly

“difficult” compounds for solvation energy modeling. Thus, users of G09 solvation models with default

settings will need to be aware that historical ΔG°solv benchmarking studies using G03 and earlier

versions (see also, e.g., [18]) are not necessarily applicable to the most recent version of this software.

We note that G09 has a “G03Defaults” option for solvent settings to allow users to reproduce default

G03 solvent model calculations as closely as possible, but perfect agreement is not always possible due

to revisions to the software between these two versions.

In general, prior ΔG°solv benchmarking studies have used DFT methods, and concluded that increasing

basis set completeness either has no significant effect on the quality of ΔG°solv predictions, or in some

cases may even decrease the predictive accuracy (see, e.g., [3, 18-22]). Consequently, we performed

calculations using the HF and B3LYP model chemistries with the larger 6-311++G(d,p) basis set, and

also added calculations at these levels using the SMD solvent model [20] in G09. To probe the effect of

density functional on the results, we also employed the M062X/6-311++G(d,p) level of theory [7-12, 23]

with the IEFPCM-UFF and SMD solvent models (Table 2). At the HF and DFT level, we see no

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substantial difference in ΔG°solv prediction accuracy with increasing level of theory (although the

IEFPCM-UFF/B3LYP/6-311++G(d,p) method reaches closer to near chemically accurate MAD/RMSD

of 1.1/1.2 kcal/mol), but we observe a significant difference in error metrics between the B3LYP and

M062X functionals at basis set equivalence using the SMD solvent model. The SMD solvent model

also does not perform as well as the IEFPCM-UFF model using any level of theory considered for

these compounds.

Calculations were then conducted using the CBS-Q//B3,[24, 25] G4MP2,[26] and G4 [27] composite methods

in G09 with the IEFPCM-UFF and SMD solvent models (Table 3). At these levels of theory, chemical

accuracy is obtained by the IEFPCM-UFF solvent model using all three methods, with G4

MAD/RMSD of 0.7/0.9 kcal/mol and no systematic error (MSD=0.0 kcal/mol). The maximum errors

for outliers are also reduced from up to 4 kcal/mol for HF and DFT levels of theory to no larger than

1.5 kcal/mol with the composite methods. While still maintaining higher ΔG°solv errors for this suite of

compounds, the SMD model errors are substantially reduced (RMSD reduction of about 33%, and loss

of the systematic low ΔG*solv bias) in moving from B3LYP/6-311++G(d,p) up to the composite

methods. The error difference between the M062X/6-311++G(d,p) and composite method calculations

is more pronounced, with the RMSD reduced by more than 50% and the elimination of a substantial

systematic underbias of 2.0 kcal/mol.

We found these results intriguing, and sought to examine whether the significant reduction in ΔG°solv

errors moving from HF/DFT to composite methods with the IEFPCM-UFF and SMD solvent models

for the subset of “difficult” neutral compounds would hold when applied against sets of “simpler”

neutral and ionic compounds often employed in previous benchmarking studies. For this, we used a set

of 18 anionic, 15 cationic, and 25 neutral inorganic and organic compounds and their corresponding

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experimental ΔG*solv data from the Gaussian 98 (G98) and G03 CPCM benchmarking study of Takano

and Houk [18] and ΔG*solv compilation of Pliego and Riveros.[28] ΔG*

solv were converted to ΔG°solv as

described above. Unlike the subset of “difficult” compounds, we find little difference in the overall

error metrics for the 25 “simpler” neutral compounds between B3LYP/6-31+G** and G4 ΔG°solv

estimates using either the IEFPCM-UFF or SMD solvent models (Table 4). However, substantial

ΔG°solv differences (>1 kcal/mol) are observed at the B3LYP/6-31+G** and G4 ΔG°solv levels for the

following two compounds (all values in kcal/mol): acetone, -3.5 (IEFPCM-UFF/B3LYP/6-31+G**),

-1.3 (IEFPCM-UFF/G4), -4.3 (SMD/B3LYP/6-31+G**), -1.3 (SMD/G4); and water, -3.8 (IEFPCM-

UFF/B3LYP/6-31+G**), -1.9 (IEFPCM-UFF/G4), -7.5 (SMD/B3LYP/6-31+G**), -5.0 (SMD/G4).

The error trend with increasing level of theory for water also reverses with a change in solvent model,

with B3LYP/6-31+G**→G4 substantially decreasing the prediction accuracy (by 1.8 kcal/mol) using

the IEFPCM-UFF model while substantially increasing the prediction accuracy (by 2.5 kcal/mol) with

SMD model. These apparently isolated anomalous error trendings with level of theory and solvent

model on simple and fundamental compounds complicate the generalizability of ΔG°solv benchmarking

efforts.

All methods have substantial difficulty with anions (MAD ~12.5 kcal/mol for IEFPCM-UFF and ~10

to 10.5 kcal/mol for SMD) and cations (MAD ~15 kcal/mol for IEFPCM-UFF and ~5 to 5.5 kcal/mol

for SMD), but display good predictive accuracy for neutral compounds (MAD<2 kcal/mol). In

particular, the SMD solvent model exhibits chemical accuracy without systematic bias for the neutral

subclass at the B3LYP/6-31+G** and G4 levels, with MSD/MAD of 0.0/0.8 and 0.9/1.0 kcal/mol,

respectively. A modest improvement (MAD reduction of ~2 kcal/mol compared to IEFPCM-UFF) is

obtained from the SMD model for anions, whereas the SMD model achieves an almost 3-fold reduction

in MAD for cations versus the IEFPCM-UFF model. There is generally good agreement between the

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two levels of theory within a particular solvent model for both anions and cations, with the exceptions

of Br- (IEFPCM-UFF), CH3COO- (SMD), HC≡C- (IEFPCM-UFF and SMD), OH- (IEFPCM-UFF and

SMD), and OOH- (IEFPCM-UFF and SMD) for which differences of >1 kcal/mol were observed. The

bulk error metrics for the ionic species also mask large individual ΔG°solv differences between the

solvation models. Although the MAD between the IEFPCM-UFF and SMD models for anions differs

by only ~2 kcal/mol across both levels of theory, the individual differences range up to 14 kcal/mol

(and up to 19 kcal/mol for the cations). We also note that as part of their CPCM benchmarking study,

Takano and Houk [18] found similarly poor performance for the UFF cavity with anions and cations at

the HF/6-31+G(d)//B3LYP/6-31+G(d) level, with the best performance (MAD~3 to 5 kcal/mol) using

the UAHF and UAKS cavities.

Finding that the IEFPCM-UFF model outperformed the SMD model for the subset of neutral “difficult”

compounds, the SMD model outperformed the IEFPCM-UFF model for the neutral “simpler”

compounds, and that level of theory effects were only clearly evident with the “difficult” compounds,

we needed an additional suite of potentially “difficult” compounds upon which to draw some overall

conclusions. A set of 25 compounds was chosen from the compendium in Marenich et al.,[21] and

calculations were conducted with the B3LYP/6-31+G** and G4 methods using the IEFPCM-UFF and

SMD solvent models (Table 5). For the 16 mono- and polyhalogenated hydrocarbons (fluoro-, chloro-,

and bromo- substituted methanes, ethanes, and ethenes) increasing the level of theory results in

significant error reductions, much as we observed with the subset of 9 “difficult” compounds from ref.

[3] as presented in Table 3. The MAD is reduced by ~50% and 25% with the SMD and IEFPCM-UFF

models, respectively, when the G4 method is used compared to the B3LYP/6-31+G** level of theory.

The systematic ΔG°solv underbias is also eliminated at the SMD/G4 level, and substantially reduced at

the IEFPCM-UFF/G4 level. No large changes in ΔG°solv (i.e., >1 kcal/mol) were observed when the

8

theoretical level was increased; rather, the overall error reduction with the G4 method generally

involved relatively small error reductions for most compounds with no clear outliers.

With the known branching errors in DFT functionals (particularly B3LYP) for relative energies of

linear and branch alkane isomers, one may expect analogous ΔG°solv errors present at the DFT level

which are reduced with composite methods. Although we only considered n-hexane and 2-

methylpentane, we know that the B3LYP branching error manifests itself already at the C6 alkane

homolog between these two isomers.[29] However, we observe no difference in ΔG°solv (or the relative

ΔG°solv; ΔΔG°solv) between these two compounds using the B3LYP/6-31+G** and G4 methods with

either solvent model. This does not rule out branching errors in ΔG°solv at higher homologs or with

more highly branched congeners, which should be systematically investigated as part of future solvent

benchmarking efforts. There is no ΔG°solv prediction improvement with increasing level of theory for

the cyclic compounds, and for cyclopentane, the G4 calculations result in a substantial decrease of

accuracy. As with nitromethane in the initial 9 compound “difficult” subset, three additional

nitroalkanes (nitroethane, 1-nitropropane, and 2-nitropropane) collectively display a significant

improvement in ΔG°solv accuracy at the G4 level as compared to the B3LYP/6-31+G** method,

particularly with the IEFPCM-UFF solvent model. These examples are sufficient to unambigously

demonstrate the possibility for HF/DFT methods to suggest problems in solvent models for some

functional groups, whereas the use of higher level methods appears to lower the perceived ΔG°solv

inaccuracies.

Overall, the findings show that the default IEFPCM-UFF and CPCM solvation models in G09 have

significantly improved ΔG°solv prediction accuracy for a subset of highly polar/polyfunctional organic

compounds relative to previous software versions. In addition, while increasing level of theory does not

9

improve ΔG°solv estimation performance for a representative standard set of neutral and ionic

monofunctional common benchmarking compounds having alkanol, amine, thiol, nitrile, aldehyde,

ketone, carboxylic acid, ester, ether, phenol, peroxide, and main group polyhydride moieties, higher

level methods appear to significantly improve the ΔG°solv prediction capability against more polar and

polyfunctional molecules containing nitro, sulfoxide, sulfonyl, and halogenated moieties. The results

are perhaps intuitive, as low levels of HF/DFT methods can obtain reasonably accurate geometries and

energies for simple organic compounds. On the other hand, a number of previous studies have shown

that high-level ab initio and composite methods are required to achieve suitable performance against

more problematic functionalities, particularly moieties such as polyhalogenated groups and highly

branched hydrocarbons.[30-45] Difficult compounds are not only troublesome from the theoretical

perspective. Often, these compounds (notably the polyhalogenated members) pose experimental

challenges for determining partitioning properties such as vapor pressure, solubility, and Henry's law

constants (from which ΔG°solv is calculated, either directly or by proxy). Thus, as benchmarking ΔG°solv

moves into difficult compounds at high levels of theory, as with other areas of computational

thermodynamics (e.g., enthalpies of formation/isomerization), we may find the high level theoretical

values are more reliable than the experimental estimates, while HF/DFT approaches do not reach the

required accuracy.

Consequently, the rationale for continuing to use only HF/DFT calculations when benchmarking

solvent models is not clear. If the goal is to isolate and benchmark the accuracy of solvation models,

particularly across the broadest suite of functional groups possible, the highest practical levels of theory

should be employed. We concede that high level calculations cannot be performed on larger molecular

and macromolecular systems for which future solvation model development is often intended. In these

cases, solvation inaccuracies will likely result from errors in both the reduced level of theory needed

10

for the computations to remain practical, as well as errors in the solvent model, and these sources will

be difficult to separate (as well as the inherently more difficult and unreliable experimental ΔG°solv

determinations for large molecules). When lower levels of theory are applied, it is not clear whether the

model chemistry, basis set, or the solvation model is at fault when discrepancies from experimental

data are encountered. Furthermore, the wide diversity of density functionals and basis sets, and likely

equal diversity of ΔG°solv prediction capabilities, also complicates the task. Such ambiguity prevents

discriminating whether the solvent model is not accurately modeling a particular functional group, or

whether the level of theory is at issue. Similarly, good agreement with experimental data using HF/DFT

approaches could result from error cancellation, thereby masking underlying fundamental problems

with the solvent model.

Since the potentially problematic moieties (out of the large set of all possible functional groups) whose

ΔG°solv accuracy is dependent on the use of high level methods cannot readily be identified in advance,

and with the advent of high performance computing platforms and the relatively cost efficient CBS-

Q//B3, G4MP2, and G4 composite methods for compounds with <10 to 15 heavy atoms (and

practicality of the W1BD method [46] for systems with <6 heavy atoms), the sole use of HF and DFT

approaches for benchmarking solvation models against neutral and ionic species should be

discontinued. Instead, high level calculations should complement HF/DFT solvent benchmarking

efforts to the greatest degree possible. These types of multiple theory level benchmarks will help the

community best identify the combination of method and solvent model suitable for the task at hand,

and better assist in the accurate parametrization of existing and proposed solvation models.

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Acknowledgements

This work was made possible by the facilities of the Western Canada Research Grid

(WestGrid:www.westgrid.ca; project 100185), the Shared Hierarchical Academic Research Computing

Network (SHARCNET:www.sharcnet.ca; project aqn-965), and Compute/Calcul Canada.

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[40] K.W. Sattelmeyer, J. Tirado-Rives, W.L. Jorgensen, J. Phys. Chem. A 110 (2006) 13551.

[41] M.D. Wodrich, C. Corminboeuf, P.V.R. Schleyer, Org. Lett. 8 (2006) 3631.

[42] P.R. Schreiner, A.A. Fokin, R.A. Pascal, A. de Meijere, Org. Lett. 8 (2006) 3635.

[43] Y. Zhao, D.G. Truhlar, Org. Lett. 8 (2006) 5753.

[44] S. Grimme, M. Steinmetz, M. Korth, J. Chem. Theory Comput. 3 (2007) 42.

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1851.

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(2009) 2687.

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15

Table 1. Comparison of experimental solvation free energies (ΔG°solv) for 9 organic compounds and corresponding theoretical estimates at the HF/6-31+G** and B3LYP/6-31+G** levels of theory using the IEFPCM and CPCM solvent models in Gaussian 03 (G03) and Gaussian 09 (G09). Experimental and G03 ΔG°solv data taken from ref. [3]. Values in brackets represent deviations from experimental ΔG°solv. Values are in kcal/mol.

HF/6-31+G** B3LYP/6-31+G**compound expt. [3] IEFPCM-UAHF-G03 [3] IEFPCM-UFF-G09 CPCM-G03 [3] CPCM-G09 IEFPCM-UAHF-G03 [3] IEFPCM-UFF-G09 CPCM-G03 [3] CPCM-G091,4-dioxane -3.2 -6.8 [-3.6] -2.6 [0.5] -6.9 [-3.7] -2.7 [0.5] -6.4 [-3.2] -2.2 [0.9] -6.4 [-3.4] -2.2 [0.9]2,2,2-trifluoroethanol -2.4 -6.7 [-4.3] -4.5 [-2.1] -6.8 [-4.4] -4.6 [-2.2] -6.3 [-3.9] -4.1 [-1.7] -6.4 [-4.0] -4.2 [-1.8]cyanobenzene -2.2 -2.5 [-0.3] -4.1 [-1.9] -2.5 [-0.3] -4.1 [-1.9] -1.6 [0.6] -3.3 [-1.1] -1.6 [0.6] -3.3 [-1.1]dimethyl sulfide 0.4 0.3 [-0.0] -0.9 [-1.2] 0.3 [-0.1] -0.9 [-1.3] 0.6 [0.3] -0.6 [-0.9] 0.6 [0.2] -0.6 [-0.9]dimethyl sulfoxide -6.8 -0.8 [6.0] -8.8 [-2.0] -0.9 [5.9] -8.8 [-2.0] 0.7 [7.5] -6.8 [0.0] 0.5 [7.3] -6.8 [0.0]methyl t-butyl ether -0.3 -0.0 [0.3] -0.8 [-0.5] -0.1 [0.2] -0.8 [-0.5] 0.1 [0.4] -0.6 [-0.3] 0.1 [0.4] -0.6 [-0.3]nitrobenzene -2.2 -2.4 [-0.2] -4.4 [-2.2] -2.5 [-0.3] -4.4 [-2.2] -1.4 [0.8] -3.6 [-1.3] -1.5 [0.7] -3.6 [-1.4]nitromethane -2.3 -5.0 [-2.7] -3.9 [-1.6] -5.0 [-2.7] -5.2 [-2.9] -3.6 [-1.4] -4.0 [-1.7] -3.6 [1.3] -4.0 [-1.7]sulfolane -6.8 -10.5 [-3.7] -10.1 [-3.3] -10.6 [-3.8] -10.2 [-3.4] -8.7 [1.9] -8.2 [-1.5] -8.7 [-2.0] -8.3 [-1.5]

MSDa -1.0 -1.6 -1.0 -1.8 -0.1 -0.5 -0.1 -0.5MADb 2.4 1.7 2.4 1.9 2.2 1.2 2.2 1.2RMSDc 3.1 1.9 3.2 2.1 3.1 1.4 3.1 1.4

a mean signed deviation. b mean absolute deviation. c root mean squared deviation.

16

Table 2. Comparison of experimental solvation free energies (ΔG°solv) for 9 organic compounds and corresponding theoretical estimates at the HF/6-31+G**, HF/6-311++G(d,p), B3LYP/6-31+G**, B3LYP/6-311++G(d,p), and M062X/6-311++G(d,p) levels of theory using the IEFPCM-UFF and SMD solvent models in Gaussian 09 (G09). Experimental ΔG°solv data taken from ref. [3]. Values in brackets represent deviations from experimental ΔG°solv. Values are in kcal/mol.

IEFPCM-UFF SMDcompound expt. [3] HF/6-311++G(d,p) B3LYP/6-311++G(d,p) M062X/6-311++G(d,p) HF/6-31+G** HF/6-311++G(d,p) B3LYP/6-31+G** B3LYP/6-311++G(d,p) M062X/6-311++G(d,p)1,4-dioxane -3.2 -2.5 [0.6] -2.2 [0.9] -2.4 [0.8] -4.8 [-1.7] -4.5 [-1.3] -4.8 [-1.7] -4.5 [-1.3] -4.9 [-1.7]2,2,2-trifluoroethanol -2.4 -4.5 [-2.1] -4.3 [-1.8] -4.3 [-1.9] -6.2 [-3.8] -6.2 [-3.8] -6.2 [-3.8] -4.6 [-2.1] -5.6 [-3.2]cyanobenzene -2.2 -4.0 [-1.8] -3.3 [-1.1] -3.3 [1.1] -3.0 [-0.8] -2.9 [-0.7] -3.0 [-0.8] -1.8 [0.4] -2.0 [0.2]dimethyl sulfide 0.4 -0.8 [-1.2] -0.5 [-0.8] -0.6 [-0.9] 0.7 [0.3] 0.7 [0.4] 0.7 [0.3] 1.0 [0.7] 0.9 [0.5]dimethyl sulfoxide -6.8 -9.1 [-2.3] -7.0 [-0.2] -7.6 [-0.7] -12.8 [-6.0] -13.4 [-6.6] -12.8 [-6.0] -10.3 [-3.5] -11.3 [-4.5]methyl t-butyl ether -0.3 -0.8 [-0.5] -0.6 [-0.3] -0.6 [-0.3] -0.3 [0.0] -0.2 [0.1] -0.3 [0.0] -1.3 [-1.0] 0.0 [0.3]nitrobenzene -2.2 -4.4 [-2.2] -3.5 [-1.3] -3.5 [-1.3] -3.1 [-0.9] -3.1 [-0.9] -3.1 [-0.9] -1.5 [0.7] -1.6 [0.6]nitromethane -2.3 -4.0 [-1.7] -4.0 [-1.7] -3.7 [-1.4] -4.0 [-1.6] -4.0 [-1.7] -4.0 [-1.6] -3.1 [-0.8] -3.7 [-1.4]sulfolane -6.8 -10.4 [-3.6] -8.4 [-1.6] -9.7 [-2.9] -15.9 [-9.1] -16.6 [-9.8] -15.9 [-9.1] -13.1 [-6.3] -16.2 [-9.4]

MSDa -1.6 -0.9 -1.1 -2.6 -2.7 -1.3 -1.5 -2.0MADb 1.8 1.1 1.3 2.7 2.8 1.9 1.9 2.4RMSDc 2.0 1.2 1.4 4.0 4.2 2.5 2.6 3.7

a mean signed deviation. b mean absolute deviation. c root mean squared deviation.

17

Table 3. Comparison of experimental solvation free energies (ΔG°solv) for 9 organic compounds and corresponding theoretical estimates at the CBS-Q//B3, G4MP2, and G4 levels of theory using the IEFPCM-UFF and SMD solvent models in Gaussian 09 (G09). Experimental ΔG°solv data taken from ref. [3]. Values in brackets represent deviations from experimental ΔG°solv. Values are in kcal/mol.

IEFPCM-UFF SMDcompound expt. [3] CBS-Q//B3 G4MP2 G4 CBS-Q//B3 G4MP2 G41,4-dioxane -3.2 -1.9 [1.3] -1.7 [1.4] -1.7 [1.4] -3.7 [-0.5] -3.4 [-0.2] -3.4 [-0.2]2,2,2-trifluoroethanol -2.4 -3.7 [-1.3] -3.3 [-0.9] -3.3 [-0.9] -3.7 [-1.3] -4.9 [-2.5] -4.9 [-2.5]cyanobenzene -2.2 -2.8 [-0.6] -2.6 [-0.4] -2.6 [-0.4] -1.1 [1.1] -0.9 [1.3] -0.9 [1.3]dimethyl sulfide 0.4 -0.2 [-0.6] -0.2 [-0.6] -0.2 [-0.6] 1.3 [0.9] 1.3 [0.9] 1.3 [0.9]dimethyl sulfoxide -6.8 -5.4 [1.4] -5.3 [1.5] -5.3 [1.5] -7.4 [-0.6] -7.5 [-0.7] -7.6 [-0.8]methyl t-butyl ether -0.3 -0.4 [-0.1] -0.3 [0.0] -0.3 [0.0] 1.0 [1.3] 1.2 [1.5] 1.2 [1.5]nitrobenzene -2.2 -2.5 [-0.3] -2.3 [-0.1] -2.3 [-0.1] 0.2 [2.4] 0.5 [2.8] 0.5 [2.7]nitromethane -2.3 -3.4 [1.1] -3.2 [-0.9] -3.2 [-0.9] -2.0 [0.3] -1.5 [0.8] -1.5 [0.8]sulfolane -6.8 -6.5 [0.3] -6.4 [0.4] -6.4 [0.4] -9.9 [-3.1] -9.6 [-2.8] -9.6 [-2.8]

MSDa -0.1 0.0 0.0 0.1 0.1 0.1MADb 0.8 0.7 0.7 1.3 1.5 1.5RMSDc 0.9 0.9 0.9 1.6 1.8 1.8

a mean signed deviation. b mean absolute deviation. c root mean squared deviation.

18

Table 4. Comparison of experimental solvation free energies (ΔG°solv) for a suite of 18 anionic, 15 cationic, and 25 neutral inorganic and organic compounds and corresponding theoretical estimates at the B3LYP/6-31+G** and G4 levels of theory using the IEFPCM-UFF and SMD solvent models in Gaussian 09 (G09). Experimental data taken from ref. [18, 28]. Values in brackets represent deviations from experimental ΔG°solv. Values are in kcal/mol.

IEFPCM-UFF SMDexpt. B3LYP/6-31+G** G4 B3LYP/6-31+G** G4

anions Br- -66.7 -61.7 [5.0] -63.1 [3.6] -50.6 [16.1] -51.3 [15.4] CH3COO- -75.4 -62.5 [12.9] -62.1 [13.3] -69.2 [6.2] -68.1 [7.3] CH3O- -93.3 -69.2 [24.1] -68.6 [24.7] -77.7 [15.6] -77.9 [15.4] CN- -65.7 -63.8 [1.9] -64.3 [1.4] -60.8 [4.9] -61.4 [4.3] Cl- -72.7 -67.1 [5.6] -68.0 [4.7] -62.6 [10.1] -63.2 [9.5] F- -103.1 -83.0 [20.1] -82.4 [20.7] -85.5 [17.6] -84.8 [18.3] HCC- -74.2 -64.7 [9.5] -66.2 [8.0] -64.8 [9.4] -66.6 [7.6] HCOO- -74.3 -63.6 [10.7] -63.3 [11.0] -68.1 [6.2] -67.3 [7.0] N3

- -68.8 -59.4 [9.4] -59.3 [9.5] -61.3 [7.5] -61.1 [7.7] OBr- -74.0 -63.5 [10.5] -63.1 [10.9] -61.6 [12.4] -60.9 [13.1] OCl- -78.8 -65.9 [12.9] -65.4 [13.4] -71.2 [7.6] -70.0 [8.8] C2H5O- -89.2 -65.7 [23.5] -65.4 [23.8] -74.1 [15.1] -74.3 [14.9] OH- -103.1 -81.0 [22.1] -78.9 [24.2] -92.9 [10.2] -90.0 [13.1] OOH- -95.4 -74.8 [20.6] -73.6 [21.8] -87.1 [8.3] -85.2 [10.2] O- -98.0 -78.7 [19.3] -77.8 [20.2] -89.0 [9.0] -88.2 [9.8] PH2

- -57.5 -59.8 [-2.3] -60.4 [-2.9] -59.2 [-1.7] -59.9 [-2.4] HS- -69.7 -64.7 [5.0] -65.1 [4.6] -60.0 [9.7] -60.3 [9.4] CH3S- -71.8 -63.4 [8.4] -63.4 [8.4] -57.5 [14.3] -57.4 [14.4]

MSDa 12.2 12.3 9.9 10.2MADb 12.4 12.6 10.1 10.5RMSDc 14.4 14.8 10.9 11.3

cations CH3CH2OH2

+ -86.5 -60.7 [25.8] -60.8 [25.7] -73.7 [12.8] -73.8 [12.7] (CH3)2C=OH+ -74.9 -56.0 [18.9] -55.9 [19.0] -65.8 [9.1] -64.8 [10.1] CH3CONH3

+ -71.9 -63.0 [8.9] -62.6 [9.3] -73.9 [-2.0] -73.0 [-1.1] (CH3)2NH2

+ -66.7 -57.9 [8.8] -57.8 [8.9] -65.6 [1.1] -65.4 [1.3] CH3NH3

+ -74.6 -63.4 [11.2] -63.6 [11.0] -72.3 [2.3] -72.2 [2.4] CH3OH2

+ -91.2 -65.1 [26.1] -65.2 [26.0] -78.9 [12.3] -78.6 [12.6] (CH3)2OH+ -77.9 -58.4 [19.5] -58.0 [19.9] -67.0 [10.9] -66.8 [11.1] (CH3)2SH+ -62.6 -55.5 [7.1] -55.3 [7.3] -57.6 [5.0] -57.3 [5.3] C2H5NH3

+ -71.1 -60.7 [10.4] -60.7 [10.4] -69.7 [1.4] -69.6 [1.5] H3O+ -108.3 -75.5 [32.8] -75.5 [32.8] -94.7 [13.6] -94.3 [14.0] HCONH3

+ -80.6 -69.3 [11.3] -68.8 [11.8] -79.8 [0.8] -78.8 [1.8] NH4

+ -83.3 -70.5 [12.8] -70.6 [12.7] -80.4 [2.9] -80.5 [2.8] C6H5NH3

+ -70.9 -56.1 [14.8] -55.8 [15.1] -65.6 [5.3] -65.1 [5.8] CH3CH2CH2NH3

+ -69.6 -58.9 [10.7] -58.8 [10.8] -68.5 [1.1] -68.3 [1.3] (pyridine)H+ -59.2 -51.1 [8.1] -51.0 [8.2] -58.0 [1.2] -57.6 [1.6]

MSD 15.1 15.3 5.2 5.5

19

MAD 15.1 15.3 5.5 5.7RMSD 16.9 17.0 7.2 7.4

neutrals C2H5OH -3.2 -1.9 [1.3] -1.3 [1.8] -3.7 [-0.5] -2.7 [0.4] (CH3)2NH -2.4 -0.5 [1.9] -0.4 [2.0] -2.0 [0.4] -1.8 [0.6] CH3CHO -1.6 -2.4 [-0.8] -1.7 [-0.1] -2.3 [-0.7] -1.0 [0.6] CH3CN -2.0 -3.7 [-1.7] -3.2 [-1.2] -2.6 [-0.6] -1.8 [0.2] CH3COCH3 -2.0 -3.5 [-1.5] -1.3 [0.6] -4.3 [-2.3] -1.3 [0.6] CH3COOH -4.8 -3.4 [1.4] -2.6 [2.2] -4.6 [0.2] -4.6 [0.2] CH3NH2 -2.7 -1.3 [1.4] -0.9 [1.8] -2.4 [0.3] -1.8 [0.9] CH3OCH3 0.0 -0.6 [-0.6] -0.2 [-0.2] 0.1 [0.1] 1.0 [1.0] CH3OH -3.2 -2.0 [1.2] -1.3 [1.9] -3.4 [-0.2] -2.3 [0.9] CH3SH 0.7 -0.6 [-1.3] -0.1 [-0.8] 0.6 [0.0] 1.1 [0.5] H2O -4.4 -3.8 [0.7] -1.9 [2.5] -7.5 [-3.1] -5.0 [-0.6] HBr -1.5 0.1 [1.6] 0.5 [2.0] 1.2 [2.7] 1.4 [2.9] HCCH 1.9 -0.3 [-2.2] 0.0 [-1.9] 1.7 [-0.2] 2.1 [0.3] HCN -1.3 -2.7 [-1.4] -2.4 [-1.0] -0.3 [1.0] 0.2 [1.5] HCOOH -5.1 -3.1 [2.0] -2.4 [2.7] -4.5 [0.6] -3.1 [2.0] HCl -0.3 -0.4 [-0.1] 0.2 [0.5] -0.1 [0.2] 0.5 [0.8] HF -5.7 -2.1 [3.5] -1.5 [4.2] -3.2 [2.5] -2.3 [3.4] HOBr -2.7 -1.6 [1.1] -0.9 [1.7] -2.4 [0.3] -1.7 [1.0] HOCl -3.8 -1.6 [2.3] -0.9 [2.9] -3.3 [0.6] -2.3 [1.6] HOOH -6.7 -2.8 [3.9] -2.2 [4.5] -7.7 [-1.0] -6.4 [0.3] NH3 -2.4 -2.1 [0.3] -1.3 [1.1] -2.6 [-0.2] -1.4 [1.0] PH3 2.9 0.9 [-1.9] 1.3 [-1.6] 2.4 [-0.5] 2.9 [0.0] C6H5OH -4.7 -2.5 [2.2] -2.0 [2.7] -4.2 [0.5] -3.3 [1.4] C6H5SH -0.7 -1.0 [-0.3] -0.6 [0.1] -0.2 [0.4] 0.3 [1.0] SH2 1.5 -0.5 [-2.0] 0.2 [-1.3] 0.0 [-1.4] 0.7 [-0.8]

MSD 0.4 1.1 0.0 0.9MAD 1.5 1.7 0.8 1.0RMSD 1.8 2.1 1.2 1.3

a mean signed deviation. b mean absolute deviation. c root mean squared deviation.

20

Table 5. Comparison of experimental solvation free energies (ΔG°solv) for a suite of 25 potentially “difficult” neutral organic compounds and corresponding theoretical estimates at the B3LYP/6-31+G** and G4 levels of theory using the IEFPCM-UFF and SMD solvent models in Gaussian 09 (G09). Experimental data taken from ref. [21, 47]. Values in brackets represent deviations from experimental ΔG°solv. Values are in kcal/mol.

IEFPCM-UFF SMDexpt. B3LYP/6-31+G** G4 B3LYP/6-31+G** G4

halocarbons fluoromethane 1.7 -0.7 [-2.3] -0.2 [-1.9] 0.7 [-1.0] 1.4 [-0.2] tetrafluoromethane 5.1 1.2 [-3.8] 1.3 [-3.7] 4.7 [-0.4] 4.9 [-0.2] chloromethane 1.3 -0.3 [-1.7] 0.1 [-1.2] 0.6 [-0.7] 1.2 [-0.2] dichloromethane 0.5 -0.8 [-1.4] -0.3 [-0.8] -0.6 [-1.2] 0.1 [-0.4] chloroform 0.8 -0.2 [-1.1] 0.2 [-0.7] 0.3 [-0.5] 0.8 [0.0] chlorofluoromethane 1.1 -1.1 [-2.2] -0.5 [-1.6] -0.5 [-1.6] 0.4 [-0.7] chlorodifluoromethane 1.4 -0.5 [-1.9] 0.0 [-1.4] 0.7 [-0.7] 1.4 [0.0] fluorotrichloromethane 2.7 1.0 [-1.8] 1.1 [-1.6] 3.4 [0.6] 3.5 [0.7] dibromomethane -0.2 -0.7 [-0.5] -0.3 [-0.1] -0.7 [-0.5] -0.3 [-0.1] tribromomethane -0.1 -0.4 [-0.3] 0.0 [0.1] -0.5 [-0.4] -0.2 [-0.2] 1-chloro-2,2,2-trifluoroethane 2.0 -1.2 [-3.2] -0.7 [-2.7] -0.1 [-2.0] 0.7 [-1.2] 1,1,1-trichloroethane 1.7 -0.2 [-1.8] 0.2 [-1.4] 0.5 [-1.2] 1.1 [-0.6] 1,1,1,2-tetrachloroethane 0.8 -0.8 [-1.5] -0.2 [-1.0] -0.4 [-1.1] 0.4 [-0.4] trichloroethene 1.5 0.4 [-1.1] 0.7 [-0.9] 2.2 [0.7] 2.5 [1.0] tetrachloroethene 2.0 0.9 [-1.0] 1.1 [-0.9] 3.6 [1.7] 3.6 [1.7] 1,1,1-trifluoropropan-2-ol -2.3 -2.6 [-0.4] -2.2 [0.1] -3.6 [-1.3] -2.6 [-0.3]

MSDa -1.6 -1.2 -0.6 -0.1MADb 1.6 1.2 1.0 0.5RMSDc 1.9 1.6 1.1 0.7

linear and branched alkanes n-hexane 4.4 1.4 [-3.0] 1.4 [-3.0] 3.4 [-1.0] 3.3 [-1.1] 2-methylpentane 4.4 1.4 [-3.0] 1.4 [-3.1] 4.0 [-0.4] 4.0 [-0.5]cyclic cyclopropane 2.7 1.1 [-1.6] 1.2 [-1.5] 1.7 [-1.0] 1.8 [-0.9] cyclopentane 3.1 1.8 [-1.3] -0.4 [-3.5] 3.3 [0.2] 1.7 [-1.4] cyclohexane 3.1 1.5 [-1.7] 1.6 [-1.5] 2.9 [-0.2] 2.9 [-0.2]

21

aziridine -3.5 -1.8 [1.7] -1.2 [2.3] -6.9 [-3.4] -6.3 [-2.8] azetidine -3.7 -0.8 [2.8] -0.6 [3.1] -4.0 [-0.4] -3.7 [0.0] tetrahydrofuran -1.6 -1.0 [0.6] -0.7 [0.8] -1.2 [0.4] -0.4 [1.2] tetrahydropyran -1.2 -0.7 [0.5] -0.4 [0.8] -1.3[-0.1] -0.8 [0.5]nitro nitroethane -1.8 -3.6 [-1.7] -2.9 [-1.1] -2.8 [-1.0] -1.4 [0.4] 1-nitropropane -1.4 -4.1 [-2.7] -2.3 [-0.9] -2.3 [-0.9] -0.4 [1.0] 2-nitropropane -1.2 -3.6 [-2.4] -3.2 [-2.0] -2.0 [-0.7] -0.4 [0.8]

a mean signed deviation. b mean absolute deviation. c root mean squared deviation.

22